Circuit Response
every model has an edge, and the figure carries it
Concept

Nyquist rate — where it appears

Twice the highest frequency a signal contains, being the lowest sample rate at which it can be represented without folding. Meeting it is a statement about the signal after the anti-alias filter rather than before it, which is what makes that filter's stopband the real specification.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A 9.0 kHz input sampled at 10 kHz arrives as 1.0 kHz. computed by solving, not by drawing. The dots are the samples. The input at 9.00 kHz is above half the 10 kHz rate, and every dot also lies on the 1.00 kHz curve drawn beside it — the two sample sequences differ by 9.3e-15, which is the arithmetic and not a small effect. Nothing is attenuated and nothing is distorted: the samples are the samples of a different signal, at full amplitude, and there is no measurement of them that could say which one was there.

The frequency a sample rate invents

Every other boundary on this site is a model getting gradually worse. This one has no gradient at all: below half the sample rate a set of samples has one sinusoid through it, above half the sample rate it has another, and the two sets of numbers are identical to three parts in ten thousand billion. Nothing is attenuated, nothing is distorted, and there is no measurement of the samples that could say which signal was there.

digital · Sample rate What the reconstruction returns, either side of 5.0 kHz. computed by solving, not by drawing. The Whittaker–Shannon sum is evaluated on the samples and compared with two things: the signal that was sampled, and the frequency the samples report. Below 5.00 kHz these are the same curve and the error is 4.88e-3 — the truncation of the sum at sixty-four samples either side, and nothing else. Above it they part: at 9.00 kHz the reconstruction is 1.59e-3 from the alias and 2.000 from the input. The small number is the interesting one. A reconstruction cannot be improved into the right answer, because it is already an exact answer to a different question.

An exact answer to a different question

The reconstruction that turns samples back into a signal is normally introduced as the thing that recovers what was there. Measured on both sides of half the sample rate it does something more interesting than failing: above the boundary it returns the alias to 1.6 parts in a thousand, which is the same accuracy it returns the input with below the boundary, and it is wrong about the input by twice the amplitude. Its error is not a degradation. It is exactness about something else.

digital · Sample rate A switched-capacitor low-pass driven past half its own clock. computed by solving, not by drawing. The clock is 1.00 MHz and the corner the rung below fitted is 1.59 kHz. The falling dashed curve is that continuous model, which knows nothing about a clock and goes on falling. The circles are the marched circuit, read at the frequency the output actually appears at. They part company past half the clock and by 992 kHz the model is 42.1 decibels wrong — an input just below the clock arrives just above direct current, in the middle of the passband, with the passband's own gain. The third curve is the exact discrete transfer function evaluated at the folded frequency, and it agrees with the march to 0.26 decibels, which is what says the march is measuring the folding rather than an artefact of itself.

The filter that samples

The rung below built a resistor out of a clock and measured two ways it is not one: a settling time, and a corner that is a capacitor ratio rather than an R–C product. Both are errors in a value and both get smaller as the design gets better. This is an error of a different kind — the arrangement is not a continuous system at all, and nothing below half the clock shows it. An input at 992 kilohertz arrives at 7.8 kilohertz with the passband's own gain, where the continuous model the rung below fitted says it is 56 decibels down.

filters · Switched capacitor

Named alongside it

The objects these essays reach for when they reach for this one.

AliasingSample rateModel rangeAnti-alias filterConvergence orderFilter tradeoffMarchingReconstructionSwitched capacitor

All concepts